Calculate the slant height for the given cone. Round to the nearest tenth.
2 answers:
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Answer:
The minimum head breadth that will fit the clientele is 4.4 inches.
The maximum head breadth that will fit the clientele is 7.8 inches.
Step-by-step explanation:
Let <em>X</em> = head breadths of men that is considered for the helmets.
The random variable <em>X</em> is normally distributed with mean, <em>μ</em> = 6.1 and standard deviation, <em>σ</em> = 1.
To compute the probability of a normal distribution we first need to convert the raw scores to <em>z</em>-scores using the formula:
It is provided that the helmets will be designed to fit all men except those with head breadths that are in the smallest 4.3% or largest 4.3%.
Compute the minimum head breadth that will fit the clientele as follows:
P (X < x) = 0.043
⇒ P (Z < z) = 0.043
The value of <em>z</em> for this probability is:
<em>z</em> = -1.717
*Use a <em>z</em>-table.
Compute the value of <em>x</em> as follows:
Thus, the minimum head breadth that will fit the clientele is 4.4 inches.
Compute the maximum head breadth that will fit the clientele as follows:
P (X > x) = 0.043
⇒ P (Z > z) = 0.043
⇒ P (Z < z) = 1 - 0.043
⇒ P (Z < z) = 0.957
The value of <em>z</em> for this probability is:
<em>z</em> = 1.717
*Use a <em>z</em>-table.
Compute the value of <em>x</em> as follows:
Thus, the maximum head breadth that will fit the clientele is 7.8 inches.
Short Answer
There are two numbers
x1 = -0.25 + 0.9682i <<<< answer 1
x2 = - 0.25 - 0.9582i <<<< answer 2
I take it there are two such numbers.
Let one number = x
Let one number = y
x + y = -0.5
y = - 0.5 - x (1)
xy = 1 (2)
Put equation 1 into equation 2
xy = 1
x(-0.5 - x) = 1
-0.5x - x^2 = 1 Subtract 1 from both sides.
-0.5x - x^2 - 1 = 0 Order these by powers
-x^2 - 0.5x -1 = 0 Multiply though by - 1
x^2 + 0.5x + 1 = 0 Use the quadratic formula to solve this.
a = 1
b = 0.5
c = 1
x = [-0.5 +/- sqrt(0.25 - 4)] / 2
x = [-0.5 +/- sqrt(-3.75)] / 2
x = [-0.25 +/- 0.9682i
x1 = -0.25 + 0.9682 i
x2 = -0.25 - 0.9682 i
These two are conjugates. They will add as x1 + x2 = -0.25 - 0.25 = - 0.50.
The complex parts cancel out. Getting them to multiply to 1 will be a little more difficult. I'll do that under the check.
Check
(-0.25 - 0.9682i)(-0.25 + 0.9682i)
Use FOIL
F:-0.25 * -0.25 = 0.0625
O: -0.25*0.9682i
I: +0.25*0.9682i
L: -0.9682i*0.9682i = - 0.9375 i^2 = 0.9375
Notice
The two middle terms (labled "O" and "I" ) cancel out. They are of opposite signs.
The final result is 0.9375 and 0.0625 add up to 1
There are two numbers
x1 = -0.25 + 0.9682i <<<< answer 1
x2 = - 0.25 - 0.9582i <<<< answer 2
I take it there are two such numbers.
Let one number = x
Let one number = y
x + y = -0.5
y = - 0.5 - x (1)
xy = 1 (2)
Put equation 1 into equation 2
xy = 1
x(-0.5 - x) = 1
-0.5x - x^2 = 1 Subtract 1 from both sides.
-0.5x - x^2 - 1 = 0 Order these by powers
-x^2 - 0.5x -1 = 0 Multiply though by - 1
x^2 + 0.5x + 1 = 0 Use the quadratic formula to solve this.
a = 1
b = 0.5
c = 1
x = [-0.5 +/- sqrt(0.25 - 4)] / 2
x = [-0.5 +/- sqrt(-3.75)] / 2
x = [-0.25 +/- 0.9682i
x1 = -0.25 + 0.9682 i
x2 = -0.25 - 0.9682 i
These two are conjugates. They will add as x1 + x2 = -0.25 - 0.25 = - 0.50.
The complex parts cancel out. Getting them to multiply to 1 will be a little more difficult. I'll do that under the check.
Check
(-0.25 - 0.9682i)(-0.25 + 0.9682i)
Use FOIL
F:-0.25 * -0.25 = 0.0625
O: -0.25*0.9682i
I: +0.25*0.9682i
L: -0.9682i*0.9682i = - 0.9375 i^2 = 0.9375
Notice
The two middle terms (labled "O" and "I" ) cancel out. They are of opposite signs.
The final result is 0.9375 and 0.0625 add up to 1